A circle C₁ passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of circle C₁. Let C₂ be the circle with OA as a diameter. If the tangent to C₂ at the point A meets the x-axis at P and y-axis at Q, then QA :AP is equal to

Consider the ellipse
$\frac{x^2}{4} + \frac{y^2}{3} = 1$.
Let $H (α, 0), 0 < α < 2$ , be a point. A straight line drawn through H parallel to y-axis crosses the ellipse and its auxiliary circle at points E and F respectively, in the first quadrant. The tangents to the ellipse at the point E intersects the positive x-axis at a point G. Suppose the straight line joining F and the origin makes an angle $\phi$ with the positive x-axis.

List-I		List-II
(I)	If $\Phi = \frac{\pi}{4}$, then the area of the triangle FGH is	P	$\frac{(\sqrt{3}-1)^4}{8}$
(II)	If $\Phi = \frac{\pi}{3}$, then the area of the triangle FGH is	Q	1
(III)	If $\Phi = \frac{\pi}{6}$, then the area of the triangle FGH is	R	$\frac{3}{4}$
(IV)	If $\Phi = \frac{\pi}{12}$, then the area of the triangle FGH is	S	$\frac{1}{2\sqrt{3}}$
		T	$\frac{3\sqrt{3}}{2}$

If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16, then |a| is equal to :

If the line x – 1 = 0 is a directrix of the hyperbola kx² – y² = 6, then the hyperbola passes through the point

If the length of the perpendicular drawn from the point P(a, 4, 2), a> 0 on the line
$\frac{x+1}{2} = \frac{y-3}{3} = \frac{z-1}{1}$ is $2\sqrt6$ units and $Q(α1, α2, α3)$
is the image of the point P in this line, then
$\alpha + \sum_{i=1}^{3} \alpha_i$
is equal to :

Let the solution curve y = f(x) of the differential equation
$\frac{dy}{dx} + \frac{xy}{x^2 - 1} = + \frac{ x^4+2x}{\sqrt{1 - x^2}}, \quad x \in (-1, 1)$ pass through the origin. Then
$\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) \,dx$is

Let $→{b}=^i+^j+λ k$ , λ ∈ R. If vector a is a vector such that → a×→ b= 13^i −^j−4^k and→ a.→ b+ 21 =0, then (→ b − → a) . ( → k − ^ j ) + ( → b + → a ) . ( ^ i − ^ k ) is equal to

Let $\vec{a}$ be a vector which is perpendicular to the vector
$3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}. $If $\vec{a}×(2\hat{i}+\hat{k})=2\hat{i}−13\hat{j}−4\hat{k}$
, then the projection of the vector on the vector
$ 2\hat{i}+2\hat{j}+\hat{k}$ is:

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then b + 3g is equal to ______.

The radius of the sphere is increased by 100% then the volume of the resultant sphere is increased by

If
$0 < x< \frac{1}{\sqrt2}\ and\ \frac{\sin^{-1}x}{α} = \frac{\cos^{-1}x}{β} $
then a value of
$sin(\frac{2πα}{α+β}) $
is

For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to

The equation of a common tangent to the parabolas y = x² and y = –(x – 2)² is

If A = {1,2,3........,10} then number of subsets of A containing only odd number is

The vector projection of b on a , where a = 3$\hat {i}$ + 2$\hat {j}$ + 5$\hat {k}$ and b = 7$\hat {i}$ - 5$\hat {j}$ - $\hat {k}$ is

List-I		List-II
(I)	If \(\Phi = \frac{\pi}{4}\), then the area of the triangle FGH is	P	\(\frac{(\sqrt{3}-1)^4}{8}\)
(II)	If \(\Phi = \frac{\pi}{3}\), then the area of the triangle FGH is	Q	1
(III)	If \(\Phi = \frac{\pi}{6}\), then the area of the triangle FGH is	R	\(\frac{3}{4}\)
(IV)	If \(\Phi = \frac{\pi}{12}\), then the area of the triangle FGH is	S	\(\frac{1}{2\sqrt{3}}\)
		T	\(\frac{3\sqrt{3}}{2}\)